Suppose that r is a double root of f (x) = 0; that is, f (r ) = f ′(r ) = 0 but f ′′(r ) ≠ 0, and suppose that f and all derivatives up to and including the second are continuous in some neighborhood of r . Show that en+1 ≈ 1/2 en for Newton’s method and thereby conclude that the rate of convergence is linear near a double root. (If the root has multiplicity m, then en+1 ≈ [(m − 1)/m]en.)
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